UNCORRECTED PROOF
Determination of flexibility factors in curved pipes with end restraints
using a semi-analytic formulation
E.M.M. Fonseca
a,*, F.J.M.Q. de Melo
b, C.A.M. Oliveira
ba
Department of Applied Mechanics, Polytechnic Institute of Braganc¸a, Campus De Sta Apolonia Apartado 134, Braganc¸a 5301 857, Portugal
b
Department of Mechanical Engineering and Industrial Management, Faculty of Engineering, University of Porto, Portugal Received 17 January 2002; revised 30 August 2002; accepted 17 September 2002
Abstract
Piping systems are structural sets used in the chemical industry, conventional or nuclear power plants and fluid transport in general-purpose process equipment. They include curved elements built as parts of toroidal thin-walled structures. The mechanical behaviour of such structural assemblies is of leading importance for satisfactory performance and safety standards of the installations. This paper presents a semi-analytic formulation based on Fourier trigonometric series for solving the pure bending problem in curved pipes. A pipe element is considered as a part of a toroidal shell. A displacement formulation pipe element was developed with Fourier series. The solution of this problem is solved from a system of differential equations using mathematical software. To build-up the solution, a simple but efficient deformation model, from a semi-membrane behaviour, was followed here, given the geometry and thin shell assumption. The flexibility factors are compared with the ASME code for some elbow dimensions adopted from ISO 1127. The stress field distribution was also calculated. q 2002 Published by Elsevier Science Ltd.
Keywords: Curved pipe; Pure bending; Flexibility factor; Semi-analytic formulation; Fourier series
1. Introduction
Pipe bends play a very important role in the global piping arrangement, as they not only allow flow direction change, but they also absorb thermal expansions and longitudinal deformations from adjacent tangent parts or other straight pipe elements [1]. Computer codes have been used extensively to expedite calculations of deflections, reactions and stresses both in piping system and pressure vessels. In order to determine such calculations, it is important to know flexibility factor values, where the case of pure bending is one of the most used in design[2]. Many experiments and theories have been presented to demonstrate and explain the flexibility and stresses of a curved pipe compared with elastic beam theory. In this paper we will present the results of flexibility factors using a formulation for a curved pipe subjected to uniform bending moment. We have determined the flexibility factor when the curved pipe was fitted with
thin or thick flanges. Throughout this formulation beam theory combined with the shell membrane equations is assumed.
2. Equation formulations 2.1. Essential assumptions
The deformation field refers to membrane strains and curvature variations.
The following assumptions [3] were considered in the present analysis:
(a) the curvature radius R is assumed much larger than the section radius r;
(b) a semi-membrane deformation model is adopted and neglects the bending stiffness along the longitudinal direction of the toroidal shell but considers the meridional bending resulting from ovalization; (c) the shell is thin and inextensible along the meridional
direction.
International Journal of Pressure Vessels and Piping xx (0000) xxx–xxx
www.elsevier.com/locate/ijpvp
* Corresponding author. Tel.: þ351-273-303-157; fax: þ351-273-313-051.
E-mail address: efonseca@ipb.pt (E.M.M. Fonseca). 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112
UNCORRECTED PROOF
Fig. 1shows the essential parameters defining the pipebend geometry.
2.2. The displacement field
For the displacement field we have superposed the effect of the rotation with trigonometric terms, for problems by symmetric bending wðxÞ ¼ x Ldaþ Xnx k$1 rksin kpx
L ðcomplete set of termsÞ ð1aÞ
wðxÞ ¼ x Ldaþ Xnx k$1 rk L ð2k 2 1Þp £ sinð2k 2 1Þpx
L ðwith odd termsÞ ð1bÞ The displacement field for the curved beam coincident with a central arch is:
WðxÞ ¼ 2ðwðxÞdx ð2Þ
UðxÞ ¼ 21 R
ð
WðxÞdx ð3Þ
Here, trigonometric terms for section ovalization are included Xnx k$1 cosð2k 2 1Þpx 2L £X nu i$2 2 aki
i sin iu ðfor v displacementÞ ð4Þ Xnx k$1 cosð2k 2 1Þpx 2L £X nu i$2
akicos iu ðfor w displacementÞ ð5Þ
Similar expressions are defined for the transverse section warping displacement field. Thus, expressions for warping u-displacement in transverse sections of the curved pipe either with rigid or thin flanges are:
Xnx k¼1 sinkpx L Xnu i$2
bkicos iu ðfor rigid flangesÞ ð6aÞ
Xnx k¼1 sinkpx 2L Xnu i$2
bkicos iu ðfor thin flangesÞ ð6bÞ
Finally, following the formulation proposed by Thomson
[6], the displacement field in a curved pipe resulting from the superposition of displacement shell and the complete Fourier expansion for ovalization and warping terms, is: u ¼ UðxÞ þ ðr cosuÞwðxÞ þX nx k¼1 sinkpx L £X nu i$2
bkicos iu; ðfor rigid flangesÞ ð7aÞ
u ¼ UðxÞ þ ðr cosuÞwðxÞ þX nx k¼1 sinkpx 2L £X nu i$2
bkicos iu; ðfor thin flangesÞ ð7bÞ
v ¼ 2WðxÞsinuþX nx k$1 Xnu i$2 2aki i cosð2k 2 1Þpx 2L ðsin iuÞ ð8Þ w ¼ þWðxÞcosuþX nx k$1 Xnu i$2 aki cos ð2k 2 1Þpx 2L ðcos iuÞ ð9Þ
Depicting the displacement field in a condensed matrix representation u v w 8 > > < > > : 9 > > = > > ; ¼ x3 6LRþ r cosu x L 0 {CSi} x2 2Lsinu {Si} 0 2x 2 2Lcosu {CSi} 0 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 da ~ r ~ a ~ b 8 > > > > > < > > > > > : 9 > > > > > = > > > > > ; ð10Þ where {CSi} ¼ Xnx k¼1 sinkpx L Xnu i$2 bkicos iuor {CSi} ¼ Xnx k¼1 sinkpx 2L Xnu i$2 bkicos iu; {Si} ¼ Xnx k$1 Xnu i$2 2aki i cosð2k 2 1Þpx 2L ðsin iuÞ and {CSi} ¼ Xnx k$1 Xnu i$2 aki cos ð2k 2 1Þpx 2L ðcos iuÞ:
2.3. The deformation field
As referred to previously, the deformation model considers that the pipe undergoes a semi-membrane strain field. The strain field is given by the following equations Fig. 1. Geometric parameters for the curved pipe.
113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224
UNCORRECTED PROOF
used by Melo, Flu¨gge and Kitching[3 – 5]
~ 1 ¼ 1x gxu Ku 8 > > < > > : 9 > > = > > ; ¼ › ›x 2 sinu R cosu R 1 r › ›u þ sinu R › ›x 0 0 2 1 r2 › ›u 1 r2 ›2 ›u2 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 u v w 8 > > < > > : 9 > > = > > ; ð11Þ
where 1x is the longitudinal membrane strain, gxu is the
shear strain and Ku is the meridional curvature from
ovalization.
The element stiffness matrix K is calculated from the matrix equation: ½K ¼ð x¼L x¼0 ðu¼2p u¼0 ½BT½D½Br dx d u ð12Þ
The integration of this equation is extended to the pipe surface, where matrix [B ] results from:
½B ¼ ½L £ ½N ð13Þ
Given that the pipe is inextensible along the meridional direction, no contribution for elastic strain energy arises from such strains. The elasticity matrix D appears with a simpler algebraic definition, having deleted the contribution of off-diagonal terms with a Poisson factor:
D ¼ Eh 1 2n2 0 0 0 Eh 2ð1 þnÞ 0 0 0 Eh 3 12ð1 2n2Þ 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 ð14Þ
where E is the elasticity modulus, h is the pipe thickness and
nis Poisson’s is ratio.
The stress fields are determined by:
~ s¼ Nxx Nxu Muu 8 > > < > > : 9 > > = > > ; ¼ ½D 1xx gxu Kuu 8 > > < > > : 9 > > = > > ; ð15Þ
The matrix force – displacement equation for the curved pipe element is ½K ~ de¼ ~ F ð16Þ
where K is the stiffness matrix anddeis a nodal unknown
displacement vector.
3. Determination of flexibility factor in curved pipes subjected to uniform bending in the curvature plane
The flexibility coefficient is a parameter that makes possible an accurate calculation for reactions at restraints and attachments in a piping system. For that task, one can use design codes either based on displacement or force methods, having included the flexibility factor as a stiffness reduction parameter for beam elements, whenever curved pipes are involved. In high temperature loading, creep phenomena can be present and a flexibility factor may be used to evaluate the time-dependent variation of the curved pipe bending resistance. The flexibility coefficient is calculated as follows
K ¼ MCðdaÞ MSðdaÞ
ð17Þ
where MCðdaÞ is a resistant moment in a curved pipe
subjected to a bending angledaat the edge and MSðdaÞ is
the corresponding bending moment in a straight pipe, with a length equal to that of the curved one and subjected to the same bending angle at the edge.
The calculation of the bending moment in a circular section can be determined by an integral equation. Considering only semi-membrane behaviour, the bending moment is as follows: MC¼ ð2p 0 sxhr 2 cosudu¼ Eh 1 2n2 ð2p 0 1xr 2 cosudu ð18Þ
In a thin-walled element only the deformation along x is considered for the calculation of the bending moment.
A similar calculation of the bending moment in the equivalent straight pipe is as follows:
MS¼
EI ð1 2n2Þ
da
L ð19Þ
The second moment of area for a thin tubular circular section is represented by:
I ¼ pr3h ð20Þ
Fig. 2. Geometry of a curved pipe with end restraints. 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336
UNCORRECTED PROOF
For the loading case of an imposed moment, the flexibility factors can be determined by the following equation
K ¼ da
da ð21Þ
where
da¼ MLð1 2n2Þ=EI ð22Þ anddain a curved pipe is calculated by Eq. (16).
4. The case study
The studied case is shown inFig. 2, representing a curved pipe with end restraints, subjected to a uniform bending moment. The geometry and the material properties are in accordance withTable 1, as inRef. [7]. Only one half of the pipe bend was analysed due to geometric and loading symmetry. The results refer to the transverse section at x ¼ 0 in the case of rigid or thin flanges as shown inFig. 2.
Figs. 3 and 4show the longitudinal stresses of a curved pipe under pure bending moment, using rigid or thin flanges. Those results obtained with our formulation are compared with the results by Melo and Castro [7], and Wilczek[8]. We have calculated the longitudinal stresses using all terms or only odd terms in the Fourier expansions for the displacements calculation, represented along the semi-section studied (0 – 1808).
When calculating the stress field for the curved pipes with rigid flanges we may use the formulation by all terms or only odd terms and will obtain the same results. But for the case studied with thin flanges, only the results obtained
with all terms are in good agreement with other references, as shown inFig. 4.
5. Stress analysis for other curved pipe geometry and end restraints
In Table 2, 12 examples of 908 elbows according ISO 1127 are presented. All cases refer to edge prescribed uniform bending moment. In the same way, only one half of the pipe bend was analysed and the results refer to the transverse section at x ¼ 0; as shown in Fig. 2.
In Figs. 5 and 6 we show the longitudinal and the meridional stresses for all curved pipes analysed. We have calculated the stresses in all pipes for this loading condition, having used either a complete formulation (all terms) or only odd terms in the Fourier expansions for the displacements evaluation. The top number of terms in the trigonometric expansions was fixed at eight. Next we present the results for a longitudinal membrane and meridional stress in curved pipe bends with end restraints using thick or thin flanges.
Fig. 3. Longitudinal stresses for a curved pipe with rigid flanges.
Fig. 4. Longitudinal stresses for a curved pipe with thin flanges.
Table 2
Geometric parameters for 908 elbows and material proprieties
D (mm) h (mm) R (mm) L (mm) E (N/mm2) 21.30 2 31.95 50.19 2.10 £ 105 33.70 2 50.55 79.40 2.10 £ 105 60.30 2 90.45 142.08 2.10 £ 105 101.60 2 152.40 239.39 2.10 £ 105 323.90 2 485.85 763.17 2.10 £ 105 406.40 3 609.60 957.56 2.10 £ 105 508.00 3 762.00 1196.95 2.10 £ 105 609.60 3 914.40 1436.34 2.10 £ 105 711.20 4 1066.80 1675.73 2.10 £ 105 812.80 4 1219.20 1915.11 2.10 £ 105 914.40 4 1371.60 2154.50 2.10 £ 105 1016.00 4 1524.00 2393.89 2.10 £ 105 Table 1
Geometric and material properties for a curved pipe
D (mm) h (mm) R (mm) L (mm) E (N/mm2) 340 1.2 1110 826.5 7.36 £ 104 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448
UNCORRECTED PROOF
Fig. 5. Longitudinal stresses at section equidistant to edges ðx ¼ 0Þ: 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560
UNCORRECTED PROOF
Fig. 5. (continued ) 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672UNCORRECTED PROOF
Fig. 5. (continued ) 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784UNCORRECTED PROOF
Fig. 6. Meridional bending stresses at section equidistant to edges (inside pipe surface). 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896
UNCORRECTED PROOF
Fig. 6. (continued ) 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008UNCORRECTED PROOF
Fig. 6. (continued ) 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120UNCORRECTED PROOF
6. Calculation of the flexibility factor
According to the ASME Code, the calculation of the flexibility factor in a curved pipe under uniform bending is determined from the following equation:
K ¼ 1:65
h2=3 ðfor flanged bendsÞ ð23aÞ
K ¼ 1:65
h ðfor unflanged bendsÞ ð23bÞ where
h ¼ hR
r2 ð24Þ
Fig. 7 shows the results obtained with the semi-analytic formulation discussed here compared with the ASME curve, considering rigid flanges for all curved pipes studied.
Fig. 8shows the results obtained when using thin flanges. In this case the results obtained will be compared with the ASME curve for unflanged bends.
7. Conclusion
The present method is a procedure for the stress field determination or the displacement field calculations Fig. 7. Flexibility factors, curved pipes with rigid flanges.
Fig. 8. Flexibility factors, curved pipes with thin flanges. 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170 1171 1172 1173 1174 1175 1176 1177 1178 1179 1180 1181 1182 1183 1184 1185 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200 1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232
UNCORRECTED PROOF
in curved pipes, having used a semi-analytic formulation. A mathematical symbolic solver carried out the problem discussed herein, where we have formulated the displacement field using Fourier series modelling the ovalization and warping condition in curved pipes. Good agreement between our stress results and the correspond-ing data from other authors was observed. The calculation of the flexibility factors has shown some discrepancies related with ASME results, especially for the case of curved pipes with rigid flanges; however, better agreement was observed when such results were compared with similar data from Thomson [6]. Never-theless, the case of thin flanges has shown better agreement with ASME results.
The present method is simple to program and easy to operate, demanding small capacity computers and avoiding a pre-processing mesh generation for the shell definition surface, an advantage for engineering designers.
References
[1] Ohtaki S. FEM analysis of pipe bends subjected to out-of-plane bending. European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS; 2000.
[2] Natarajan R, Mirza S. Effect of internal pressure on flexibility factors in pipe elbows with end constraints. J Pressure Vessel Technol 1985;107: 60 – 3.
[3] Madureira L, Melo FQ. A hybrid formulation in the stress analysis of curved pipes. Engng Comput 2000;17(8):970 – 80.
[4] Flugge W. Thin elastic shells. Berlin: Springer; 1973.
[5] Kitching R. Smooth and mitred pipe bends. In: Gill SS, editor. The stress analysis of pressure vessels and pressure. Oxford: Pergamon Press; 1970, chapter 7.
[6] Thomson G. In plane bending of smooth pipe bends. PhD Thesis. University of Strathclyde, Scotland, UK; 1980.
[7] Melo FJMQ, Castro PMST. A reduced integration Mindlin beam element for linear elastic stress analysis of curved pipes under generalized in-plane loading. Comput Struct 1992;43(4):787 – 94. [8] Wilczek E. Statische Berechnung eines Rohrkru¨mmers mit Realen
Randbedingungen. PhD Thesis. Technischen Hochschule Aachen, Aachen; 1984. 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327 1328 1329 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344
